13 Morphological Modelling and Simulation …
437
is that only ideal crystals can be described. More complex crystal shapes that may
result from abrasion or agglomeration cannot be described. Therefore, alternative
descriptions are explained here.
In mathematical terms, facetted convex shapes consisting of points x in space can
be described by H-representations [11]:
C(h) = {x|Ax ≤ h}
(1)
where A is a matrix with normal vectors of facets in space a i , and h is a vector with the
length of these vectors. The H-representation describes all points that are within the
facets defined by A and h. For crystals, rows of A can be interpreted as normal vectors
of the faces of such a crystal. A is therefore specific for each crystal morphology.
The vector h then describes the exact shape and size of an ideal crystal of form A.
Herein, the dimension of x is either two or three dimensional (2D/3D).
For crystals it is common that symmetry conditions apply. To reduce the dimension
of h, one can, therefore, introduce a crystal model-specific group mapping matrix
M h C →h [11]
h = M h C →h · h C
(2)
This operation can also be formulated in a reverse way with the pseudo inverse
matrix M
+
h C →h [11]
h C = M
+
h C →h · h
(3)
These constrained crystal models are called constrained H C -representations
[11–13].
Because H-representations only allow the description of ideal single crystals,
the representation is extended so that also more complex shapes can be described.
Abraded crystals, for instance, may not have sharp edges that would be described by
an H-representation, but may appear with round edges and corners. Agglomerates on
the other hand may be concave and can therefore also not be represented by simple
H-representations.
For the description of rounded particles, an ideal kernel crystal (index k) was
combined with a sphere. The combination of sets in general can be performed by the
so-called Minkowski addition [14]. The combination of i sets S 1...i can be written as
S =
S i =
x i |x i ∈ S i ∀i
.
(4)
The addition of an ideal kernel crystal C(h k ) and a sphere B that has the radius
λ r [11] is accordingly written as
C(h k , λ r ) = C(h k ) + λ r B.
(5)
437
is that only ideal crystals can be described. More complex crystal shapes that may
result from abrasion or agglomeration cannot be described. Therefore, alternative
descriptions are explained here.
In mathematical terms, facetted convex shapes consisting of points x in space can
be described by H-representations [11]:
C(h) = {x|Ax ≤ h}
(1)
where A is a matrix with normal vectors of facets in space a i , and h is a vector with the
length of these vectors. The H-representation describes all points that are within the
facets defined by A and h. For crystals, rows of A can be interpreted as normal vectors
of the faces of such a crystal. A is therefore specific for each crystal morphology.
The vector h then describes the exact shape and size of an ideal crystal of form A.
Herein, the dimension of x is either two or three dimensional (2D/3D).
For crystals it is common that symmetry conditions apply. To reduce the dimension
of h, one can, therefore, introduce a crystal model-specific group mapping matrix
M h C →h [11]
h = M h C →h · h C
(2)
This operation can also be formulated in a reverse way with the pseudo inverse
matrix M
+
h C →h [11]
h C = M
+
h C →h · h
(3)
These constrained crystal models are called constrained H C -representations
[11–13].
Because H-representations only allow the description of ideal single crystals,
the representation is extended so that also more complex shapes can be described.
Abraded crystals, for instance, may not have sharp edges that would be described by
an H-representation, but may appear with round edges and corners. Agglomerates on
the other hand may be concave and can therefore also not be represented by simple
H-representations.
For the description of rounded particles, an ideal kernel crystal (index k) was
combined with a sphere. The combination of sets in general can be performed by the
so-called Minkowski addition [14]. The combination of i sets S 1...i can be written as
S =
S i =
x i |x i ∈ S i ∀i
.
(4)
The addition of an ideal kernel crystal C(h k ) and a sphere B that has the radius
λ r [11] is accordingly written as
C(h k , λ r ) = C(h k ) + λ r B.
(5)